Apparatus for calculating heat exchanger performance



C. F. KAYAN Aug. 13, 1957 APPARATUS FOR CALCULATING HEAT EXCHANGERPERFORMANCE Filed Aug. 17, 1951 2 Sheets-Sheet 1 FIG. 2

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United States Patent O APPARATUS FOR CALCULATING HEAT EXCHANGERPERFORMANCE Carl F. Kayan, New York, N. Y. Application August 17, 1951,Serial No. 242,231 2 Claims. (Cl. 235--61) This invention relates to animproved method and apparatus for calculating the performance ofindustrial heat exchangers for transferring heat from'one fluid toanother.

The calculation of the performance of heat exchangers by the methods nowin use is laboriousand time-consuming, and such calculations present aserious problem when it is necessary to take into account varyingconditions Within the heat exchanger, such as varying specific heat andchanging over-all thermal conductance. Varying over-all conductance is aparticular problem when the heat transfer process involves radiation.Such would be the case, for example, if one of the fluids weresuperheated steam in turbulent flow where the conductance on the steamside of the exchanger consists in part of convection and in part ofgaseous radiation, and wherein the radiation conductance is distinctlyresponsive to temperature conditions.

In accordance with the invention an electrical circuit is arranged tosimulate the conditions existing within the heat exchanger. In order todo this the electrical circuit comprises a network of electricalresistances which are made proportional to the several thermalresistances (the inverse of thermal conductance) involved in the heatexchange process. The over-all electrical potential difference (voltagedifference) of the circuit corresponds to the over-all temperaturedifference of the heat exchanger, and differences in electricalpotential between different parts of the electrical circuit representditferences in temperature between corresponding parts of the heatexchanger. The electrical current resulting from the over-all potentialdifference, or voltage, impressed upon the circuit represents theover-all heat flow of the heat exchanger.

The thermal resistances of the heat exchanger comprise the resistance tothe warming up (heat absorption) of the cold fluid which is a functionof the specific heat of that fluid, the resistance to the cooling off(heat releasing) of the hot fluid, a function of the specific heat ofthat fluid, and the thermal resistance of the heat exchanger structure.The last includes the total series resistance to heat flow betweensource and receiver fluids. According to the present invention a methodof calculating heat exchanger performance has been provided whichincludes the step of representing the heat absorbing capacity of thecold fluid and the heat releasing capacity of the hot fluid each bymeans of a resistance to the flow of electrical current. The thermalresistance of the heat exchanger structure is also similarlyrepresented.

The invention will be better understood from the consideration of theaccompanying drawings and the following description in greater detail.In these drawings 'Fig. 1 is a diagrammatic view of a counterflowfluidto-fluid heat exchanger;

Fig. 2 is a diagram representing the variations in temperature of thehot and cold fluids as they travel through the heat exchanger;

Fig. 3 is a diagram of an electrical apparatus having a circuit insimulation of the heat exchanger of Fig. l, or a section of suchexchanger;

Fig. 4 is an illustrative development of Fig. 3;

Fig. 5 is a further illustrative development;

Fig. 6 is a diagram of an apparatus similar to that of Fig. 3, butwhichsimulates the heat exchanger of Fig. 5 divided into five sections;

Fig. 7 is a modification of Fig. 6; and i Fig. 8 is like Fig. 2, butproduced by calculation employing the apparatus of Figs. 6 or 7.

The invention will be described and discussed in connection with thecalculations for the counterflow type of heat exchanger as that typeintroduces more problems than the parallel flow type. Referring to Fig.l, the hot or heat source fluid enters the heat exchanger at 10 andtravels toward the right through the heat exchanger shell 11, leaving byconnection 12. The cold or heat absorbing .fluid travels from right toleft through the tube bundle,

which is represented by the single tube .13.

The conventional relationships for a two-fluid heat exchanger are givenherewith:

=AUAt 1 representing the transfer of heat from a hot source fluid to acold receiver fluid;

representing the heat involved in the temperature change of the coldfluid;

representing the heat involved in the temperature change of the hotfluid;

q=w c At =E At where q=time-rate of gross heat exchange, B. t. u./hr.

A=total exchanger area, ft.

U=over-all heat transfer conductance between source and receiver fluids,B. t. u./ (hr. sq. ft. per F.)

At =l igarithmie mean temperature difference (ideally),

h in c out) 1.4 h ou c in with t and t representing hot and coldsubstance temperatures, distinguished as to in and out conditions Thetemperature relationships may follows:

also be expressed as h ou n in where e =the base of natural logarithmsAnalysis of steady-state heat flow through complex structures has beenaccomplished extensively by means of electrical analogy (1, 2, 3) Thefield of usefulness for the electrical analogy approach may be greatlyextended through application to heat exchanger steady-state proc- WhereThe concept of equivalent thermal resistance may similarly be applied tothe heat exchanger fluids themselves. V i

Thus, Equations 2 and 3 may be rewritten:

where v a Rc=1/Ec, 'F/(B. t. 11.

=thermal equivalent corresponding to the ,cold sub- L stance waterequivalent, representing the resistance to the warming-up operationRh=l/Eh, F./(B. t. u. hr.)

=thermal resistance corresponding to the hot substance water equivalent,representing the resistance to the cooling-off operation Thus, inaccordance with this extension of the conventional heat transferresistance concept, Equation 4 may now be shown to include heat exchangeresistance:

q=Atm/(R/A.)=Atc/Rc=Ath/Rn (9) Therefore I v .Atm=q(R/A) (10) Atc=qRc 11Ath=qRn An electrical network maybe set up to simulate heat transfer ina fluid-to-fluid heat exchanger, illustratively, the counterflow heatexchanger indicated in Fig. 1. Herein electrical resistances are chosenproportional to thermal resistances represented in Equations l0, l1, and12, and electrical potentials are taken as representative oftemperatures. To facilitate operations, the over-all temperaturedifference (th in 'tc in) which is represented by an over-"all voltagediflerence, is considered as 100 percent; thus intermediate potentialdiiferences are indicated as some fraction of the over-all 100 percent.The equivalent over-all heat flow thus may be determined by means of thenet thermal resistance represented by the net electrical resistance ofthe simulatingcircuit. Total current flow on the. basis of the over-allimpressed direct current voltage may, on the basis of, appropriatecircuit scale factors, indicate the over-all heat flow rate. Thus theelectrical analogy analysis will permit the prediction of thermalperformance,' but, it should be noted,flonly in terms of the assumedbasic data. It is recognized that calculations could likewise bedirectly carried out on the basis of the corresponding thermalresistance circuit, but, as will be evident later, electricalexperimental operations often are more readily executed. I

. For the purposes of simplicity, the countercurrent heat exchanger ofFig. 1 has been'considered as divided into five equal-area sections. Itwill be apparent that any number of sections to suit the desiredaccuracy may be employed, i. e., the pattern of the circuit equationsmakes it possible to set them up for n sections. Fig. 2 indicates thefive sections.

' For any one'section, linear change in temperature is assumed, andconsequently,'linear. change in electrical potential in the equivalentelectrical circuit. The temperatu'redifference 'at' the midpoint, of thesection s representative of the heat flow through the section(arithmetic mean temperature difference), that is, as due to the thermalresistance for the section of area A/5 for a system of five sections andgross area A. Thus the contribution to total heat flow for this sectionis given by This heat flow is likewise represented in the change intemperatures for both the-hot andthe cold fluids. The change intemperature for the fluids will be given by Equations 11, Al'c=(Aq)Rc,and 1 2, Atn =(Aq)Rh. Circuitwise, these relationships may beincorporated into a network arrangement as given in Fig. 3. 7

Since the total change in the fluid temperatures for the section isdesired, and since the heat transfer energy coincides with the heatexchange energy for the fluids, the circuit may be modified as given inFig. 4, shown for thermal conditions, and Fig. 5, for electricalconditions. At represents the over-all difference in temperature for onesection, and Ae the corresponding overall difference in electricalpotential for the simulating electrical analogy circuit.

Fig. 4- thus represents thebasic element in the translation to anelectrical analogy circuit; Here Ac, represents the over-all potentialdifference, RH: and Re the equivalent electrical resistancecorrespondingto the heat exchange resistances Ril/Eh 'and Ra l/En, andR0 represents the total overall resistance of the'circuit. If Rurepresents the equivalent electrical resistance .corresponding to theheat transfer net thermal. resistance R/A=1/ (AU), then if n sectionsare used, the equivalent electrical resistance corresponding to thesection heat transfer resistance will be nRU, i. e., fora five-sectionsystem, SRU.v Thus, Ro=5Ru+'(RH+Rc)/2. Potential differences Ae and Aerepresent, to scale, the temperature changes for the hot and coldfluids, respectively.

In Figs. 3, 4, 5, 6 and 7 a source of adjustable electrical potential isindicated at 14 and an ammeter at 15. Also a slide-wire potentiometer isshown at 16 together with agalvanometer 17 and a-probing contact 18. Bymeans of this device the relative potential of any desired point in theresistance network can be quickly determined.

If both the heat transfer and the heat exchange fluid thermalresistances are regarded as constant over the entirearea'of theexchanger :(constant U and constant fluid specific heats), the'five-section electrical network for theheat exchangersimulation may beshown as in Fl j lhe diflere'nt electrical resistances Ry and representballast resistances to compensate for the successive and additive fluidpotential differences accomplished in each section, such'that the totalfluid temperature changes will be simulated by Am and Aec for the hotand cold fluids, respectively. The total flow is representedby the totalcircuit current I, representing the sum of the separate currentcontributions from each of the five sections. The dotted lines in thefigure, running from section to section, connect points of equalpotential, in accordance with the additive Ae principle cited above. 7 V

In accordance with the stipulation of constant U and constant fluidspecific heats, the following conditions are inforce:

Values for Ry and Rx for the different sections are given by thefollowing relationships, in which.

Ballast resistances R Ballast resistances Rx:

Fig. 6 has been calculated. tion has been assumed: Ec

EH=1000 B. t. u./(hr. F.), AU:

i. e., combining A and are as follows: Rc=200 ohms, RH:

relationships of Equatwork corresponding to The following basic informa-1000 B. t. u./(hr. F.), U. Proportional electrical values 100 ohms,Ru==100 ohms, Ro=5(100) 100-1-200) /2=650 ohms. The circuit values aregiven in Table 1:

Table 1 Section 1 2 3 4. 5

1, 636 1, 341 1, 098 see Relative potential values for each section arecalculated and shown in Table 2, indicative of the fluid temperatures. dcold fluids are 1.00 and 0,

The end values of the hot an respectively.

Table 2 H- Ae iz/Ae over-all (1-AeH)/Ae over-2.111.000.

200 0.222 0 (end) 349 The combined resistance of the network is 258ohms.

The circuit of Fig. 6 may circuit (Fig. 7) in which Rx are eliminated.The valu fled so that they will in turn as to potential drop as before.

joined to the adjoining sectio R0 and RH beyond RC1 and R115,respectively,

those of their own sections. A

current in addition to separate resistance element RF be replaced by anequivalent the ballast resistances Ry and es of R0 and RH aremodiproduce the identical results us. The

Each section is electrically circuit elements of carry extra Ran-Rm)will be Iidentified (Fig. 7) as herewith indicated: R.F=Ro

It is to be note basically constant, and thereforeRF1=RF2=RF3=RF4==RF5=RF Values for the modified ver be computed from 13and 14.

RH5'=RH, i. e., original value d that Ru, R0 and RH are On the basis ofthe preceding equations, a new network corresponding to Fig. 7 was setup. The values for this network, based on the data as used for theballast network case reported in Tables 1 and 2, are shown in Apparatuscorresponding to Fig. 7 as well as apparatus corresponding to Fig. 6have been constructed and used in heat exchanger calculations. Theexperimental results obtained from electrical measurements on thenetwork of Fig. 7, adjusted as to values of RE and Re, follow in Table4, and are to be compared with those cited in Table 2. It is to be notedthat the agreement is acceptable.

Table 4 Section H 1 2 3 4 5 Asa/Ac over-all 0.770 0.673 0.556 0.4050.218 0(end) 1-(Aeg5Ae over-all 1.00 0.952 0.891 0.817 0.724 .606

The combined resistance of the network measured 257 ohms.

Corresponding values for the temperatures of the hot and cold fluids,calculated by use of the apparatus of the invention and based onconventional algebraic heat exchanger relationships for the sameconditions as represented by Equation 5, are shown in Table 5.

Table 5 Section 1 2 3 4 5 Aea/Ae over-all 0.775 0.676 0. 550 0.405 0.22700 (end) 1EAeg)/Ae over-all 1.000 0.951 0.889 0.815 0.726 0.612

f It is to be noted that Tables 2, 4, and 5 show good agreement, itbeing recognized that Tables 2 and 4 are derived from a lumped circuitarrangement. Fig. 8 represents the results of Table 4 on a temperaturebasis.

Values for Figs. 6 and 7 (Tables 1 and 3) were set up in terms ofconstancy of values for R0 and RH, as well as for RU, for all of thedifierent sections. It is, however, to be recognized that thecorresponding thermal values for R0, RH and R (i. e., U), are themselvesa function of temperature conditions, and hence, with changes indicatedin temperatures, not constant. Since, for example, R=1/ U, and is, asindicated, composed of the hot-fluid component, the wall component, andthe cold-fluid component, R will be subject to the variations in thesecomponent resistances as dependent on temperature. Thus hFh=hc+hr (20)where hrn=composite boundary conductance, B. t. u./(hr. sq.

ft. F.)

hc=boundary conductance due to convection (forced),

B. t. u./(hr. sq. ft. F.)

h;;equivalent conductance of steam gaseous radiation corresponding tothe existent temperature conditions, ivy in B. t. u./ (hr .;sq. ft. F.)"I Accordingly, hm and Rm (=1/hFh) will be particularly subject to thetemperature conditions, hence variable throughout the traverse of thearea of the heat exchanger. R: and Rh in turn depend on the realizedvalues of specific heat, i. e., the existing temperatures. Theelectrical analogy approach lends itself especially to the solution ofthe heat exchanger problem with varying properties, such as for example,introduced through the variation of the gaseous radiation cited above,etc. Accordingly, the solution of the five-section counter-current heatexchanger problem has been developed to deal with the varyingproperties, using the electrical analogy approach. 7 Based on theballast circuit of Fig. 6, values for Rx and R throughout thefive-section circuit have been set up, noting that RmRc, and R arenolonger constant, but are considered to change from section to section.Let it be noted that Ro1=5RUi+ (Rc1+RH1)/2, Ro2=5RU2+ (RC2+RH2)/2, etc.

Values for P and Q change as the sections are traversed. The values forM and N change fromsection to section. Thus M1 RO1R'H1,' Mz'=Ro2-Rri2,etc. N1'= Rel-RC1; N2=Rv2-Rcz, etc.

The ballast resistance values for Rx and Ry at the difierent sectionsare shown In the case of the circuits designed for variable properties,the procedure is to start with estimated valuesfor the diiferentproperties, and on determinationof resultant temperatures on this basis,to correct the different values accordingly. Repeated corrections can beintro.-

duced until the accuracy requirement is satisfied.

It is to be noted that if RH=0, the resultant solution represents theheat exchanger case of constant temperature source, such as isrepresented by the ideal condenser analogy.

78 or feed water heater case. If Rc= 0, the resultant solutionrepresents the heat exchanger case of constant temperature receiver,such as is found in arefrigeration brine cooler. It is believed thattheindicated method of electrical analogy can likewisebe successfullyapplied to nonorthodox cases, suchascross-flow exchangers.

The following specific example is illustrative of the application of themethod and apparatusof the invention. Let us assume that 1100 lb./hr. ofliquid, specific heat 0.91, are to be cooled from an initial temperatureof F., by brine, 400 lb./hr., initial temperature=0 F., specific heat:1.25, in a countercurrent heat exchanger of 10 sq. ft. heating surfacearea, for which U the coetficient of heat transfer, under the flowcircumstances, is 100. (a) For constant specific heats and constantvalue of U, show the final outlet temperatures of the two fluids, thevariation of the fluid temperature throughout the exchanger, andthetotal amount of heat transferred per hour. (b) What would be theoutlet temperatures for 200 F. initial hot fluid temperature, and 60 F.initial cold fluid temperature. V

.- Here (as shown in column 5, paragraph 1).

A five-section network may be set up to represent the heat flow in theexchanger by the method of electrical These represent the relative basicresistances.

In accordance with the analogy relationships of Equations 15 and 16, thefive section network corresponding to Fig. 6 has been calculated. Forconvenience in use of resistor values, the different resistances shownabove are represented in ohms, as multiplied by a' factor of 100,000.Thus, with the relative values retained, proportional electrical valuesare as follows: Rn=100 ohms; Rc=200 ohms; Ru: 100 ohms, and Ro=5(100)+(100+200)/2= 650 ohms.

The five-section circuit values are given in Table 1. The

measured overall resistance of the network constructed in accordanceWith-the values of Table 1 measures 258 ohms. By means of the potentialdividing slide-wire device 16 and galvanorneter 17, therelativepotential at various points in the network with respect to theover-all (100%) may be determined. These values are shown in Table 2.

The thermal resistance corresponding to the measured over-all resistanceof 258 ohms is 258/ 100,000=.00258.

For100 volts over-all potential corresponding to a temperaturedifierence of 1000=100 F., the current flow shows 1388amps.corresponding to 38,800 B. t. u./hr. For volts, corresponding to anover-all temperature difference of 20060=140 F., the current flow shows.542 amps. corresponding to 54,200 B. t. u./ hr.

The temperature of the fluids at any point may be established from themeasured relative potential values obtained from the circuit. Thus, thetemperature may be calculated tFlu1d=fcold, inltial-i-CAtover-all whereC =relative potcntial value, with C=0 taken at the low or cold side ofthe circuit, and Atover-al1=difi6leI1CQ between incoming hot fluid andinitial co ld fluid, tcold, initial. 7

Thus, based on Table 2 and t, 1main=0 F., the outgoing cold fluidtemperature=tc, orif=0+.775 100=77.5 F. The outgoing hot fluidtemperature'=0+0.611 100= 61.1 F. p

In a similar fashion, the temperatures of the fluids at the diflerentsections may readily be calculated.

Fig. 7 shows the plot of fluid temperatures as calculated above.

The equivalent values for the second case with 140 F. over-alltemperature difierence are readily calculated. Thus, the outgoing coldfluid temperature will be: tnmm=60+.775 140=60+108.2=168.2 F. Theoutgoing hot fluid temperature=60+.61l 140=145.5 F.

in place of the five-section network as used above and corresponding toFig. 6, the equivalent electrical circuit corresponding to Fig. 7 may beused with similar results to the above. In addition, although these twoprevious networks of Figs. 6 and 7 are predicated on constant values ofRn, Re and Ro, since average specific heats are dependent on thetemperature ranges encountered, and the over-all conductance U maychange with temperature conditions, the network circuit valuescorresponding to Fig. 6 may be modified on the basis of Equations 23 and24, and corresponding to Fig. 7, by Equations 25 and 26.

These modifications permit the analogy network calculater of theinvention to provide analytical results which would be extremelydiflicult to establish by orthodox calculation methods. For example, thecalculations just described require only a few minutes, and if it isdesired to ascertain the effect of changes in specific heat or of theheat exchanger conductance, re-calculation using the new values of thesefactors can be done in a similarly short space of time.

I claim:

1. In an apparatus for calculating the performance of a heat exchangertransferring heat between two fluids which are continuously changing intemperature as they pass through the heat exchanger but wherein, at anygiven point in the apparatus, the temperatures of the fluids and theintervening separating wall are substantially constant, a plurality ofelectrical circuits each representing a section of said heat exchanger,each of said circuits comprising a plurality of electrical resistances,said resistances corresponding respectively to the warming-up of thecold fluid passing through said exchanger, the structural thermalresistance of the heat exchanger and the thermal resistance to thecooling-E of the hot fluid passing through 10 said exchanger, theresistances in each of said circuits being arranged in seriesconnection, each of said resistances being adjustable, said plurality ofcircuits being connected in parallel so as to cause the electricalcurrents flowing in them to be added together and thus represent theheat exchanger as a whole, means for measuring the total current flowingin said parallel circuits so as to determine the total heat. flowingbetween said fluids, and an adjustable potential source for impressingon said parallel circuits an electrical potential corresponding to theover-all temperature difierence in the particular heat exchanger to becalculated.

2. Heat exchanger calculating apparatus as set forth in claim 1 whereinmeans are provided for determining the electrical potential of any pointin any of said circuits as an indication of the temperature of thecorresponding part of the heat exchanger.

Reterences Cited in the file of this patent UNITED STATES PATENTS1,206,968 Wilsey Dec. 5, 1916 2,040,086 Goodwillie May 12, 19362,519,615 Wannamaker Aug. 22, 1950 2,598,267 Kayan May 27, 19522,630,968 Muskat Mar. 10, 1953 FOREIGN PATENTS 605,822 Great BritainJuly 30, 1948 252,807 Switzerland Oct. 16, 1948 OTHER REFERENCESHeat-Transfer Problems Solved With Roomful of R-C Networks, Electronics,April 1943; pp. 181-183.

The Theory of Mathematical Machines, F. J. Murray, revised edition,Kings Grown Press, New York (pp. III- 17 relied upon).

The Accuracy of Measurements in Lumped R-C Cable Circuits as Used in theStudy of Transient Heat Flow (Paschkis and Heisler), AIEE Transactions,volume 63, April 1944, page 165.

